Sockets for the turn signals should have 2 small "buttons" like you describe. One is for the + parking lights and the other is for + turn signal. The rest of the housing needs to be grounded to complete the circuit. If you are missing one of the wires or if the "button" got pulled out it's not going to work. Depending on the specific damage you may need to replace the whole socket.
When one of the bulbs in the flasher circuit isn't working (either burnt or bad connection) the other one on that side will flash fast. It's to let you know something is wrong. If the bulb is dim, it's usually a bad ground, or the bulb is getting old and has buildup on the glass that acts like tint and makes it dark.
My suggestion is see if you can find another socket that works and give it a shot. Good luck!